A Hölder continuity result for a class of obstacle problems under non standard growth conditions

2011 ◽  
Vol 284 (11-12) ◽  
pp. 1404-1434 ◽  
Author(s):  
Michela Eleuteri ◽  
Jens Habermann
Keyword(s):  
Hölder Continuity ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Holder Continuity ◽  
Continuity Result

The Hölder continuity for the gradient of solutions to one-sided obstacle problems

1992 ◽  
Vol 13 (9) ◽  
pp. 841-850
Author(s):  
Wang Xiang-dong ◽  
Liang Xi-ting
Keyword(s):  
Hölder Continuity ◽  
Obstacle Problems ◽  
Holder Continuity

scholarly journals Weighted Hölder continuity of Riemann-Liouville fractional integrals – Application to regularity of solutions to fractional cauchy problems with Carathéodory dynamics

2019 ◽  
Vol 22 (3) ◽  
pp. 722-749 ◽  
Author(s):  
Loïc Bourdin
Keyword(s):  
Hölder Continuity ◽  
Fractional Integrals ◽  
Regularity Of Solutions ◽  
Cauchy Problems ◽  
Holder Continuity ◽  
Integrable Functions ◽  
Continuity Result

Abstract This paper is dedicated to several original (weighted) Hölder continuity results for Riemann-Liouville fractional integrals of weighted integrable functions. As an application, we prove a new weighted continuity result for solutions to nonlinear Riemann-Liouville fractional Cauchy problems with Carathéodory dynamics.

Hölder continuity up to the boundary for a class of fractional obstacle problems

2016 ◽  
Vol 27 (3) ◽  
pp. 355-367 ◽  
Author(s):  
Janne Korvenpää ◽  
Tuomo Kuusi ◽  
Giampiero Palatucci
Keyword(s):  
Hölder Continuity ◽  
Obstacle Problems ◽  
Holder Continuity

scholarly journals Global higher integrability for minimisers of convex functionals with (p,q)-growth

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Lukas Koch
Keyword(s):  
Hölder Continuity ◽  
Growth Conditions ◽  
Holder Continuity ◽  
Higher Integrability ◽  
Convex Functionals ◽  
Global Higher Integrability ◽  
Continuity Assumption

AbstractWe prove global $$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) -regularity for minimisers of convex functionals of the form $${\mathscr {F}}(u)=\int _{\varOmega } F(x,Du)\,{\mathrm{d}}x$$ F ( u ) = ∫ Ω F ( x , D u ) d x .$$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and controlled (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n .

scholarly journals Pointwise regularity for solutions of double obstacle problems on metric spaces

2011 ◽  
Vol 109 (2) ◽  
pp. 185 ◽  
Author(s):  
Zohra Farnana
Keyword(s):  
Metric Spaces ◽  
Hölder Continuity ◽  
Obstacle Problems ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Pointwise Regularity

We study continuity at a given point for solutions of double obstacle problems. We obtain pointwise continuity of the solutions for discontinuous obstacles. We also show Hölder continuity for solutions of the double obstacle problems if the obstacles are Hölder continuous.

\mathfrak{B}_{1} classes of De Giorgi, Ladyzhenskaya, and Ural'tseva and their application to elliptic and parabolic equations with nonstandard growth

2019 ◽  
Vol 16 (3) ◽  
pp. 403-447
Author(s):  
Igor Skrypnik ◽  
Mykhailo Voitovych
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Hölder Continuity ◽  
Growth Conditions ◽  
Holder Continuity ◽  
Nonstandard Growth ◽  
Elliptic And Parabolic Equations ◽  
Functional Classes ◽  
Nonstandard Growth Conditions ◽  
Quasilinear Elliptic

The article provides an application of the generalized De Giorgi functional classes to the proof of the Hölder continuity of weak solutions to quasilinear elliptic and parabolic equations with nonstandard growth conditions.

scholarly journals Regularity results for a class of obstacle problems with p, q−growth conditions

2021 ◽  
Vol 27 ◽  
pp. 19 ◽  
Author(s):  
M. Caselli ◽  
M. Eleuteri ◽  
A. Passarelli di Napoli
Keyword(s):  
Principal Part ◽  
Lipschitz Continuity ◽  
Sobolev Class ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Local Boundedness ◽  
Bounded Set ◽  
Continuity Result ◽  
Regularity Results ◽  
Admissible Functions

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type min{ ∫ΩF(x, Dz) : z ∈ 𝛫ψ(Ω)}. Here 𝛫ψ(Ω) is the set of admissible functions z ∈ u0 + W1,p(Ω) for a given u0 ∈ W1,p(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝn, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.

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